most powerful test using Neyman-Pearson lemma with two parameters

Let denote a random sample from a population having a Poisson distribution with mean . Let denote an independent random sample from a population having a Poisson distribution with mean . Derive the most powerful test for testing versus .

According to the likelihood function,

.

So by Neyman-Pearson, our rejection region for the most powerful test is given by

.

Substituting , we have the rejection region

,

where we define by

,

and is the desired probability of a type I error, that is, the "level" of the test.

Assuming I have not made any errors in the above analysis, then I am curious, can I do anything more towards defining ? In particular, is there any way to define as a function of ?